These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.

Exercise1.1.9:A spaceship is traveling at the speed \(2t^2 + 1\)km/s(\(t\)is time in seconds). It ispointing directly away from earth and at time \(t = 0\)it is 1000 kilometers from earth. How far fromearth is it at one minute from time \(t =0\)?

Exercise1.1.104:Sid is in a car traveling at speed \(10t + 70\)miles per hour away from Las Vegas,where \(t\)is in hours. At \(t =0\)the Sid is 10 miles away from Vegas. How far from Vegas is Sid 2hours later?

Exercise1.1.105:Solve \(y' = y''\), \(y(0) = 1\), where \(n\) is a positive integer. Hint: You have toconsider different cases.

Exercise1.3.11:Find an explicit solution for \( y' = ye^{-x^2}\), \(y(0) = 1\). It is alright to leave a definiteintegral in your answer.

Exercise1.3.12:Suppose a cup of coffee is at 100 degrees Celsius at time \(t =0\), it is at 70 degreesat \(t = 10\)minutes, and it is at 50 degrees at \(t = 20\) minutes. Compute the ambient temperature.

Exercise1.3.106:TakeExercise1.3.3 with the same numbers: 89 degrees at \( t =0\), 85 degreesat \(t =1\), and ambient temperature of 22 degrees. Suppose these temperatures were measured withprecision of \( \pm 0.5\)degrees. Given this imprecision, the time it takes the coffee to cool to (exactly)60 degrees is also only known in a certain range. Find this range. Hint: Think about what kind oferror makes the cooling time longer and what shorter.

Exercise1.4.9:Suppose there are two lakes located on a stream. Clean water flows into the firstlake, then the water from the first lake flows into the second lake, and then water from the secondlake flows further downstream. The in and out flow from each lake is 500 liters per hour. The firstlake contains 100 thousand liters of water and the second lake contains 200 thousand liters ofwater. A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water isbeing continually mixed perfectly by the stream.

a) Find the concentration of toxic substance as afunction of time in both lakes.

b) When will the concentration in the first lake be below 0.001 kgper liter?

c) When will the concentration in the second lake be maximal?

Exercise1.4.10:Newton’s law of coolingstates that \(\frac{dx}{dt} = -k(x-A)\) where \(x\)is the temperature, \(t\)is time, \(A\) is the ambient temperature, and \(k>0\) is a constant. Suppose that \(A=A_0 \cos(\omega t)\) for some constants \(A_0\) and \(\omega\). That is, the ambient temperature oscillates (for example night andday temperatures). a) Find the general solution. b) In the long term, will the initial conditions makemuch of a difference? Why or why not?

Exercise1.4.11:Initially 5 grams of salt are dissolved in 20 liters of water. Brine withconcentration of salt 2 grams of salt per liter is added at a rate of 3 liters a minute. The tank ismixed well and is drained at 3 liters a minute. How long does the process have to continue untilthere are 20 grams of salt in the tank?

Exercise1.4.12:Initially a tank contains 10 liters of pure water. Brine of unknown (but constant)concentration of salt is flowing in at 1 liter per minute. The water is mixed well and drained at 1liter per minute. In 20 minutes there are 15 grams of salt in the tank. What is the concentration ofsalt in the incoming brine?

Exercise1.4.103:Suppose a water tank is being pumped out at 3L/min. The water tank startsat 10 L of clean water. Water with toxic substance is flowing into the tank at 2L/min, withconcentration \(20t\)g/Lat time \(t\). When the tank is half empty, how many grams of toxic substanceare in the tank (assuming perfect mixing)?

Exercise1.4.104:Suppose we have bacteria on a plate and suppose that we are slowlyadding a toxic substance such that the rate of growth is slowing down. That is, suppose that \(\frac{dP}{dt}=(2-0.1\, t)P\). If \(P(0)=1000\) , find the population at \(t=5\).

2There are several things called Bernoulli equations, this is just one of them. The Bernoullis were aprominent Swiss family of mathematicians. These particular equations are named forJacob Bernoulli(1654 –1705).

Exercise1.6.6:Start with the logistic equation \(\frac{dx}{dt} = kx (M -x)\). Suppose that we modify ourharvesting. That is we will only harvest an amount proportional to current population. In otherwords we harvest \(hx\)per unit of time for some \(h > 0\)(Similar to earlier example with \(h\)replacedwith \(hx\)). a) Construct the differential equation. b) Show that if \(kM > h\), then the equation is stilllogistic. c) What happens when \(kM < h\)?

Exercise1.6.103:Assume that a population of fish in a lake satisfies\(\frac{dx}{dt} = kx (M -x)\). Nowsuppose that fish are continually added at \(A\)fish per unit of time. a) Find the differential equationfor \(x\). b) What is the new limiting population?[1]

3The unstable points that have one of the arrows pointing towards the critical point are sometimes calledsemistable.

Exercise1.7.6:Example of numerical instability: Take \(y' = -5y\), \(y(0) = 1\). We know that thesolution should decay to zero as \(x\)grows. Using Euler’s method, start with \(h =1\)and compute \(y_1, y_2, y_3, y_4\)to try to approximate \(y(4)\). What happened? Now halve the interval. Keep halvingthe interval and approximating \(y(4)\)until the numbers you are getting start to stabilize (that is,until they start going towards zero). Note: You might want to use a calculator.

The simplest method used in practice is the Runge-Kutta method. Consider \(\frac{dy}{dx} = f(x, y)\), \(y(x_0) = y_0\) and a step size \(h\). Everything is the same as in Euler’s method, except the computation of \(y_{i+1}\) and \(x_{i+1}\).

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